\documentclass[reqno]{amsart}
\usepackage{hyperref}
\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2010(2010), No. 96, pp. 1--11.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2010 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2010/96\hfil Existence and concentration of solutions]
{Existence and concentration of solutions for a
$p$-laplace equation with potentials in $\mathbb{R}^N$}
\author[M. Wu, Z. Yang\hfil EJDE-2010/96\hfilneg]
{Mingzhu Wu, Zuodong Yang} % in alphabetical order
\address{Mingzhu Wu \newline
Institute of Mathematics, School of Mathematical Science,
Nanjing Normal University,
Jiangsu Nanjing 210046, China}
\email{wumingzhu\_2010@163.com}
\address{Zuodong Yang \newline
Institute of Mathematics, School of
Mathematical Science, Nanjing Normal University,
Jiangsu Nanjing 210046, China. \newline
College of Zhongbei, Nanjing Normal University,
Jiangsu Nanjing 210046, China}
\email{zdyang\_jin@263.net}
\thanks{Submitted January 22, 2010. Published July 15, 2010.}
\thanks{Supported by grants 10871060 from the National Natural
Science Foundation of China, \hfill\break\indent and 08KJB110005
from the Natural Science Foundation of the Jiangsu Higher
Education \hfill\break\indent Institutions of China}
\subjclass[2000]{35J25, 35J60}
\keywords{Potentials; critical point theory; concentration;
existence; \hfill\break\indent concentration-compactness; $p$-Laplace}
\begin{abstract}
We study the $p$-Laplace equation with Potentials
$$
-\operatorname{div}(|\nabla u|^{p-2}\nabla u)+\lambda
V(x)|u|^{p-2}u=|u|^{q-2}u,
$$
$u\in W^{1,p}(\mathbb{R}^N)$, $x\in \mathbb{R}^N$ where
$2\leq p$, $p0$ such that the set ${\{x\in
\mathbb{R}^N: V(x)p$ and $p^{*}=\infty$ if $1\leq N\leq p$.
\end{itemize}
Note that if $\varepsilon^{p}=\lambda^{-1}$, then $u$ is a solution
of \eqref{e1.2} if and only if $v=\lambda^{\frac{-1}{q-p}}u$ is
a solution of \eqref{e1.3}, hence as far as the existence and the
number of solutions
are concerned, these two problems are equivalent.
$\|u\|_{p}$ will denote the usual $L^{p}(\mathbb{R}^N)$ norm and
$V^{\pm}(x)=\max{\{\pm V(x),0}\}$. $B_{\rho}$ and $S_{\rho}$ will
respectively denote the open ball and the sphere of radius $\rho$
and center at the origin.
It is well known that the functional
$$
\Phi(u)={\frac{1}{p}}\int_{\mathbb{R}^N}(|\nabla
u|^{p}+V(x)|u|^{p})dx-{\frac{1}{q}}\int_{\mathbb{R}^N}|u|^{q}dx
$$
is of class $C^{1}$ in the Sobolev space
\begin{equation}
E={\{u\in W^{1,p}(\mathbb{R}^N):\|u\|^{p}=\int_{\mathbb{R}^N}(|\nabla
u|^{p}+V^{+}(x)|u|^{p})dx2$, using
the lower semi-continuity of the $L^{p}$-norm with respect to the
weak convergence and $u_n\rightharpoonup u$ in $W^{1,p}(\Omega)$,
we deduce
$$
\lim_{n\to\infty}\langle|\nabla u_n|^{p-2}{\nabla u_n},
{\nabla u_n}\rangle\geq \langle|\nabla u|^{p-2}{\nabla u}, {\nabla
u}\rangle
$$
and
\begin{align*}
&\lim_{n\to\infty}\langle |{\nabla u_n}-\nabla
u|^{p-2}({\nabla u_n}-{\nabla u}),{\nabla u_n}-{\nabla
u}\rangle\\
&=0 \geq \lim_{n\to\infty}\langle |{\nabla u_n}-\nabla
u|^{p-2}({\nabla u}-{\nabla u}),{\nabla u}-{\nabla u}\rangle.
\end{align*}
So
\begin{align*}
\lim_{n\to\infty}\langle|{\nabla u_n}-\nabla u|^{p-2}{\nabla
u_n},{\nabla u_n}\rangle
&\geq \lim_{n\to\infty}\langle|{\nabla u_n}-\nabla
u|^{p-2}{\nabla u_n},{\nabla u}\rangle \\
&=\lim_{n\to\infty}\langle|{\nabla u_n}-\nabla
u|^{p-2}{\nabla u},{\nabla u_n}\rangle\\
&=\lim_{n\to\infty}\langle|{\nabla u_n}-\nabla
u|^{p-2}{\nabla u},{\nabla u}\rangle.
\end{align*}
Then
\begin{align*}
&\lim_{n\to\infty}\int_{\Omega}(|\nabla u_{n}|^{p}-|\nabla
u|^{p})dx\\
& =\lim_{n\to\infty}\int_{\Omega}|\nabla
u_{n}|^{p-2}(|\nabla u_{n}|^{2}-|\nabla
u|^{2})dx+\lim_{n\to\infty}\int_{\Omega}(|\nabla
u_{n}|^{p-2}-|\nabla u|^{p-2})|\nabla u|^{2}dx\\
&=\lim_{n\to\infty}\int_{\Omega}(|\nabla
u_{n}|^{p-2}+|\nabla u|^{p-2})(|\nabla u_{n}|^{2}-|\nabla
u|^{2})dx\\
&\quad +\lim_{n\to\infty}\int_{\Omega}(|\nabla
u_{n}|^{p-2}|\nabla u|^{2}-|\nabla u|^{p-2}|\nabla
u_{n}|^{2})dx.
\end{align*}
From $u_n\rightharpoonup u$ in $W^{1,p}(\Omega)$,
$$
\lim_{n\to\infty}\int_{\Omega}(|\nabla
u_{n}|^{p-2}|\nabla u|^{2}-|\nabla u|^{p-2}|\nabla
u_{n}|^{2})dx=0.
$$
So
\begin{align*}
\lim_{n\to\infty}\int_{\Omega}(|\nabla u_{n}|^{p}-|\nabla
u|^{p})dx
&=\lim_{n\to\infty}\int_{\Omega}(|\nabla
u_{n}|^{p-2}+|\nabla u|^{p-2})(|\nabla u_{n}|^{2}-|\nabla
u|^{2})dx\\
& \geq \lim_{n\to\infty}\int_{\Omega}|\nabla
u_{n}-\nabla u|^{p-2}(|\nabla u_{n}|^{2}-|\nabla u|^{2}).
\end{align*}
So we have
\begin{align*}
&\lim_{n\to\infty}\langle|{\nabla u_n}|^{p-2}{\nabla
u_n},{\nabla u_n}\rangle+\langle|{\nabla u_n}-\nabla
u|^{p-2}{\nabla u},{\nabla u_n}\rangle+\langle|{\nabla u_n}-\nabla
u|^{p-2}{\nabla
u_n},{\nabla u}\rangle\\
&\geq \lim_{n\to\infty}\langle|{\nabla u_n}-\nabla
u|^{p-2}{\nabla u_n},{\nabla u_n}\rangle\\
&\quad +\lim_{n\to\infty}\langle|{\nabla u_n}-\nabla
u|^{p-2}{\nabla u},{\nabla u}\rangle+\langle|{\nabla
u}|^{p-2}{\nabla u},{\nabla u}\rangle.
\end{align*}
Then
\begin{align*}
&\lim_{n\to\infty}\langle|{\nabla u_n}|^{p-2}{\nabla
u_n},{\nabla u_n}\rangle \\
&\geq
\lim_{n\to\infty}\langle|{\nabla u_n}-\nabla
u|^{p-2}{{\nabla u_n} -{\nabla u}},{\nabla u_n}-{\nabla u}\rangle
+\langle|{\nabla u}|^{p-2}{\nabla u},{\nabla u}\rangle.
\end{align*}
and
$$
\lim_{n\to\infty}\int_{\Omega}|\nabla u_n|^{p}dx
\geq \lim_{n\to\infty}\int_{\Omega}|\nabla u_n-{\nabla
u}|^{p}dx+\int_{\Omega}|\nabla u|^{p}dx.
$$
For $p>3$, there exist a $k\in N$ that $00$. Suppose $\|u_m\|\to\infty$ as
$m\to\infty$ and let $w_m=u_m/\|u_m\|$. Dividing \eqref{e2.5} by
$\|u_m\|^{q}$ we see that $w_m\to 0$ in
$L^{q}(\mathbb{R}^N)$ as $m\to\infty$ and therefore
$w_m\rightharpoonup 0$ in $E$ as $m\to\infty$ after passing
to a subsequence. Hence
$\int_{\mathbb{R}^N}V^{-}(x)|w_m|^{p}dx\to 0$ as
$m\to\infty$. So dividing \eqref{e2.6} by $\|u_m\|^{p}$, it
follows that $w_m\to 0$ in $E$ as $m\to\infty$, a
contradiction. Thus ${\{u_m}\}$ is bounded.
As in the preceding proof, we may assume $u_m\rightharpoonup u$ in
$E$ and $u_m\to u$ in $L^{p}_{\rm loc}(\mathbb{R}^N)$. Set
$u_m=v_m+u$. Since $\Phi'(u)=0$ and
$$
\Phi(u)=\Phi(u)-{\frac{1}{p}}\langle\Phi'(u),u\rangle
=({\frac{1}{p}}-{\frac{1}{q}})\|u\|^{q}_{q}\geq 0,
$$
it follows from \eqref{e2.2}, \eqref{e2.3} that
$$
\lim_{m\to\infty}(|\|v_m\|^{p}-\|v_m\|^{q}_{q}|)\leq
\lim_{m\to\infty}(|\|u_m\|^{p}-\|u_m\|^{q}_{q}|
+|\|u\|^{p}-\|u\|^{q}_{q}|)=0
$$
so
\begin{equation}
\lim_{m\to\infty}(\|v_m\|^{p}-\|v_m\|^{q}_{q})=0 \label{e2.7}
\end{equation}
and
\begin{equation}
c=\lim_{m\to\infty}\Phi(u_m)
\geq \lim_{m\to\infty}(\Phi(v_m)+\Phi(u))
\geq \lim_{m\to\infty}\Phi(v_m). \label{e2.8}
\end{equation}
By \eqref{e2.7}, we have
\begin{equation}
\lim_{m\to\infty}\int_{\mathbb{R}^N}(|\nabla
v_m|^{p}+V(x)|v_m|^{p})dx=\lim_{m\to\infty}
\int_{\mathbb{R}^N}|v_m|^{q}dx=\gamma \label{e2.9}
\end{equation}
possibly after passing to a subsequence, and therefore it follows
from \eqref{e2.8} that
\begin{equation}
c\geq ({\frac{1}{p}}-{\frac{1}{q}})\gamma. \label{e2.10}
\end{equation}
By \eqref{e2.4},
$$
\lim_{m\to\infty}\int_{\mathbb{R}^N}(|\nabla
v_m|^{p}+V_{b}(x)|v_m|^{p})dx
=\lim_{m\to\infty}\int_{\mathbb{R}^N}(|\nabla
v_m|^{p}+V(x)|v_m|^{p})dx=\gamma\,.
$$
On the other hand,
$$
\|v_m\|^{p}_{q}\leq M^{-}_{b}\lim_{m\to\infty}\int_{\mathbb{R}^N}(|\nabla
v_m|^{p}+V_{b}(x)|v_m|^{p})dx;
$$
therefore, $\gamma^{\frac{p}{q}}\leq
M^{-}_{b}\gamma$. Combining this with \eqref{e2.10}, we see that either
$\gamma=0$, or
$$
c\geq ({\frac{1}{p}}-{\frac{1}{q}})M_{b}^{\frac{q}{(q-p)}}
$$
hence $\gamma$ must be $0$ by the assumption on $c$. So according
to \eqref{e2.9}, we have
$$
\lim_{m\to\infty}\int_{\mathbb{R}^N}(|\nabla
v_m|^{p}+V^{+}(x)|v_m|^{p})dx
=\lim_{m\to\infty}\int_{\mathbb{R}^N}(|\nabla
v_m|^{p}+V(x)|v_m|^{p})dx=0.
$$
Therefore, $v_m\to 0$ and
$u_m\to u$ in $E$ as $m\to\infty$.
\end{proof}
Next we recall a usual critical point theory which will be used in
the below Theorem. Here $\gamma(A)$ is the Krasnoselskii genus of
$A$.
\begin{theorem} \label{thmA}
Suppose $E\in C^{1}(M)$ is an even functional on a
complete symmetric $C^{1,1}$-manifold $M\subset V\setminus {\{0}\}$
in some Banach space $V$. Also suppose $E$ satisfies $(PS)$ and is
bounded below on $M$. Let
$\widetilde{\gamma}(M)=\sup\{\gamma(K);
K\subset M \text{ and symmetric}\}$.
Then the functional $E$ possesses at least
$\widetilde{\gamma}(M)\leq \infty$
pairs of critical points.
\end{theorem}
\section{Proof of Main Theorems}
\begin{theorem} \label{thm1}
Suppose Assumptions {\rm (V1), (P1)} are satisfied,
$\sigma(-\Delta_{p}+V)\subset (0,\infty)$,
$\sup_{x\in \mathbb{R}^N}V(x)=b>0$ and the measure of the set
${\{x\in \mathbb{R}^N:V(x)0$. Then the infimum in \eqref{e1.5} is attained at some
$u\geq 0$. If $V\geq 0$, then $u>0$ in $\mathbb{R}^N$.
\end{theorem}
\begin{proof}
Since $V^{+}$ is bounded, $E=W^{1,p}(\mathbb{R}^N)$
here. Let $u_b$ be the radially symmetric positive solution of the
equation
$$
-\operatorname{div}(|\nabla u|^{p-2}\nabla u)
+b|u|^{p-2}u=|u|^{q-2}u,\quad x\in \mathbb{R}^N.
$$
It is well known that such $u_b$ exists, is
unique and minimizes
\begin{equation}
N_b=\inf_{u\in E\setminus {\{0}\}}{\frac{\int_{\mathbb{R}^N}(|\nabla
u|^{p}+b|u|^{p})dx}{\|u\|^{p}_{q}}} \label{e3.1}
\end{equation}
(see \cite{c1}). So if $V\equiv b$, the proof is complete.
Otherwise we may assume without loss of generality that $V(0)0$. A simple computation shows that
if $\lambda>0$,
then $N_{\lambda b}$ is attained at
$$
u_{\lambda b}(x)=\lambda^{\frac{1}{(q-p)}}u_{b}
(\lambda^{\frac{1}{p}}x)\quad \mbox{and}\quad
N_{\lambda b}=\lambda^{r}N_b,
$$
where $r=1-{\frac{N}{p}}+{\frac{N}{q}}$.
Choosing $\lambda=(b-\varepsilon)/b$ we see that
$N_{b-\varepsilon}0$ in $\mathbb{R}^N$.
\end{proof}
\begin{theorem} \label{thm2}
Suppose $V\geq 0$ and {\rm (V1), (V2), (P1)} are
satisfied. Then there exists $\Lambda>0$ such that for each
$\lambda\geq \Lambda$ the infimum in \eqref{e1.5} is attained at some
$u_{\lambda}>0$. Here $V(x)$ replaced by $\lambda V(x)$.
\end{theorem}
\begin{proof}
Here $V=V^{+}$. Let $b$ be as in (V2) and
\begin{equation}
\begin{gathered}
M^{\lambda}=\inf_{u\in E\setminus {\{0}\}}{\frac{\int_{\mathbb{R}^N}(|\nabla
u|^{p}+\lambda V(x)|u|^{p})dx}{\|u\|^{p}_{q}}},\\
M^{\lambda}_{b}=\inf_{u\in E\setminus {\{0}\}}{\frac{\int_{\mathbb{R}^N}(|\nabla
u|^{p}+\lambda
V_{b}(x)|u|^{p})dx}{\|u\|^{p}_{q}}}.
\end{gathered}\label{e3.3}
\end{equation}
It suffices to show that $M^{\lambda}0$ so that $V(x)0$,
\begin{align*}
M^{\lambda}
&\leq {\frac{\int_{\mathbb{R}^N}(|\nabla w_{\lambda
b}|^{p}+\lambda V(x)|w_{\lambda b}|^{p})dx}{\|w_{\lambda
b}\|^{p}_{q}}}\\
& \leq {\frac{\int_{\mathbb{R}^N}(|\nabla w_{\lambda
b}|^{p}+\lambda (b-\varepsilon)|w_{\lambda
b}|^{p})dx}{\|w_{\lambda b}\|^{p}_{q}}}\\
&\leq \lambda^{r}({\frac{\int_{\mathbb{R}^N}(|\nabla u_b|^{p}
+\lambda b|u_b|^{p})dx-\varepsilon\int_{\mathbb{R}^N}|u_b|^{p}dx}
{\|u_b\|^{p}_{q}}}+\varepsilon)\\
& \leq\lambda^{r}(N_b-C_{0}\varepsilon)
\end{align*}
where $N_b$ is defined in
\eqref{e3.1} and $r$ in \eqref{e3.2}. Using \eqref{e3.2}
and \eqref{e3.3} we also see that
\begin{equation}
M^{\lambda}_{b}\geq \inf_{u\in E\setminus
{\{0}\}}{\frac{\int_{\mathbb{R}^N}(|\nabla u|^{p}+\lambda
b|u|^{p})dx}{\|u\|^{p}_{q}}}=N_{\lambda
b}=\lambda^{r}N_b, \label{e3.4}
\end{equation}
hence
$M^{\lambda}0$.
\end{proof}
Next we consider the existence of multiple solutions under the
hypothesis that $V^{-1}(0)$ has nonempty interior.
\begin{theorem} \label{thm3}
Suppose $V\geq 0$, $V^{-1}(0)$ has nonempty
interior and {\rm (V1), (V2), (P1)} are satisfied. For each $k\geq
1$ there exists $\Lambda_{k}>0$ such that if $\lambda\geq
\Lambda_{k}$, then \eqref{e1.2} has at the least $k$ pairs of nontrivial
solutions in $E$.
\end{theorem}
\begin{proof}
For a fixed $k$ we can find
$\varphi_1$,\dots,$\varphi_{k}\in C^{\infty}_{0}(\mathbb{R}^N)$ such
that $\operatorname{supp}\varphi_{j}$, $1\leq j\leq k$, is contained in
the interior of $V^{-1}(0)$ and
$\operatorname{supp}\varphi_{i}\cap{\operatorname{supp}\varphi_{j}}
=\emptyset$
whenever $i\neq j$. Let
$$
F_{k}=\operatorname{span}{\{\varphi_{1},\dots,\varphi_{k}}\}.
$$
Since $V\geq 0$,
$\Phi(u)={\frac{1}{p}}\|u\|^{p}-{\frac{1}{q}}\|u\|^{q}_{q}$ and
therefore there exist $\alpha,\quad \rho>0$ such that
$\Phi|_{S_{\rho}}\geq \alpha$. Denote the set of all symmetric (in
the sense that $-A=A$) and closed subsets of $E$ by $\Sigma$, for each
$A\in \Sigma$ let $\gamma(A)$ be the Krasnoselski genus and
$$
i(A)=\min_{h\in \Gamma}\gamma(h(A)\cap S_{\rho})
$$
where $\Gamma$ is the set of all odd homeomorphisms $h\in C(E,E)$.
Then $i$ is a version of Benci's pseudoindex. Let
$$
\Phi_{\lambda}(u)={\frac{1}{p}}\int_{\mathbb{R}^N}(|\nabla
u|^{p}+\lambda
V(x)|u|^{p})dx-{\frac{1}{q}}\int_{\mathbb{R}^N}|u|^{q}dx,\quad
\lambda\geq 1
$$
and
$$
c_{j}=\inf_{i(A)\geq j}\sup_{u\in A}\Phi_{\lambda}(u),\quad 1\leq
j\leq k.
$$
Since $\Phi_{\lambda}(u)\geq \Phi(u)\geq \alpha$ for
all $u\in S_{\rho}$ and since $\mbox{i}(F_k)=\dim {F_{k}}=k$,
$$
\alpha\leq c_1\leq \dots \leq c_k\leq \sup_{u\in
F_k}\Phi_{\lambda}(u)=C.
$$
It is clear that $C$ depends on $k$ but
not on $\lambda$. As in \eqref{e3.4}, we have
$$
M^{\lambda}_{b}\geq N_{\lambda b}=\lambda^{r}N_{b}
$$
where $r>0$, and therefore $M^{\lambda}_{b}\to\infty$. Hence
$C0$, then, up to a
subsequence, $u_m\to\overline{u}$ in
$L^{q}(\mathbb{R}^N)$, where $\overline{u}$ is a weak solution of
the equation
\begin{equation}
-\operatorname{div}(|\nabla u|^{p-2}\nabla
u)=|u|^{q-2}u,\quad x\in\Omega, \label{e3.6}
\end{equation}
and $\overline{u}=0$ a.e. in $\mathbb{R}^N\setminus V^{-1}(0)$. If
moreover $V\geq 0$, then $u_m\to\overline{u}$ in $E$ as
$m\to\infty$.
\end{theorem}
\begin{proof}
Since $\lambda_{m}\geq 1$, $\|u_m\|\leq
\|u_m\|_{\lambda_m}\leq C$. Passing to a subsequence,
$u_m\rightharpoonup\overline{u}$ in $E$ and
$u_m\to\overline{u}$ in $L^{q}_{\rm loc}(\mathbb{R}^N)$ as
$m\to\infty$. Since
$\langle{\Phi_{\lambda_m}}'(u_m),\varphi\rangle=0$, we see that
$$
\lim_{m\to\infty}\int_{\mathbb{R}^N}V(x)|u_m|^{p-2}u_{m}\varphi dx
= 0,\quad
\int_{\mathbb{R}^N}V(x)|\overline{u}|^{p-2}\overline{u}\varphi
dx=0
$$
and for all $\varphi\in C^{\infty}_{0}(\mathbb{R}^N)$.
Therefore, $\overline{u}=0$ a.e. in $\mathbb{R}^N\setminus
V^{-1}(0)$.
We claim that $u_m\to \overline{u}$ in
$L^{q}(\mathbb{R}^N)$ as $m\to\infty$. Assuming the
contrary, it follows from Lion vanishing lemma that
$$
\int_{B_{\rho}(x_m)}|u_m-\overline{u}|^{p}dx\geq \gamma
$$
for some $\{x_m\}\subset \mathbb{R}^N$, $\rho$, $\gamma>0$ and
almost all $m$, where $B_{\rho}(x)$ denotes the open ball of
radius $\rho$ and center $x$.
Since $u_m\to\overline{u}$ in $L^{q}_{\rm loc}(\mathbb{R}^N)$,
$|x_m|\to\infty$. Therefore, the measure of the set
$B_{\rho}(x_m)\cap{\{x\in \mathbb{R}^N:V(x)0$, it follows that
$$
\|u_m\|^{p}\leq \|u_m\|^{p}_{\lambda_m}=\|u_m\|^{q}_{q}
$$
and
$$
\|\overline{u}\|^{p}= \|\overline{u}\|^{p}_{\lambda_m}
=\|\overline{u}\|^{q}_{q}.
$$
Hence $\limsup_{m\to\infty}\|u_m\|^{p}\leq
\|\overline{u}\|^{q}_{q}=\|\overline{u}\|^{p}$; therefore,
$u_m\to\overline{u}$ in $E$ as $m\to\infty$.
\end{proof}
\begin{theorem} \label{thm5}
Suppose {\rm (V1), (V2), (P1)} are satisfied and
$V^{-1}(0)$ has nonempty interior, $V\geq 0$, $u_m\in E$ is a
solution of \eqref{e3.5}, $\lambda_{m}\to\infty$ and
$\Phi_{\lambda_m}(u_m)$ is bounded and bounded away from $0$. Then
the conclusion of Theorem \ref{thm4} is satisfied and $\overline{u}\neq 0$.
\end{theorem}
\begin{proof} We have
$$
\Phi_{\lambda_m}(u_m)={\frac{1}{p}}\|u_m\|^{p}_{\lambda_m}
-{\frac{1}{q}}\|u_m\|^{q}_{q}
$$
and
$$
\Phi_{\lambda_m}(u_m)=\Phi_{\lambda_m}(u_m)
-{\frac{1}{p}}\langle{\Phi_{\lambda_m}'(u_m),u_m}\rangle
=({\frac{1}{p}}-{\frac{1}{q}})\|u_m\|^{q}_{q}
$$
Hence $\|u_m\|_{q}$, and therefore also $\|u_m\|_{\lambda_m}$ is
bounded. So the conclusion of Theorem \ref{thm4} holds. Moreover, as
$\|u_m\|_{q}$ is bounded away from $0$, $\overline{u}\neq 0$.
\end{proof}
As a consequence of this corollary, if $k$ is fixed, then any
sequence of solutions $u_m$ of \eqref{e1.2} with
$\lambda=\lambda_{m}\to\infty$ obtained in Theorem \ref{thm3}
contains a subsequence concentrating at some $\overline{u}\neq 0$.
Moreover, it is possible to obtain a positive solution for each
$\lambda$, either via Theorem \ref{thm1} or by the mountain pass theorem. It
follows that each sequence ${\{u_m}\}$ of such solutions with
$\lambda_{m}\to\infty$ has a subsequence concentrating at
some $\overline{u}$ which is positive in $\Omega$. Corresponding to
$u_m$ are solutions $v_m=\varepsilon^{p/(q-p)}_{m}u_{m}$
of \eqref{e1.3},
where $\varepsilon^{p}_{m}=\lambda^{-1}_{m}$. Then $v_m\to
0$ and $\varepsilon^{-p/(q-p)}_{m}v_{m}\to\overline{u}$.
subsection*{Remark} In the proof of Lemmas \ref{lem2} and \ref{lem3}
and Theorems \ref{thm1}--\ref{thm3}, the
condition (V1) can be replaced by
\begin{itemize}
\item[(V1')] $v\in L^{1}_{\rm loc}(\mathbb{R}^N)$ and
$V^{-}=\max{\{-V,0}\}\in L^{q}(\mathbb{R}^N)$,
where $q=N/p$ if $N\geq p+1$, $q>1$ if $N=p$ and $q=1$ if $N